Counting Cubic Extensions with given Quadratic Resolvent

Given a number field $k$ and a quadratic extension $K_2$, we give an explicit asymptotic formula for the number of isomorphism classes of cubic extensions of $k$ whose Galois closure contains $K_2$ as quadratic subextension, ordered by the norm of their relative discriminant ideal. The main tool is Kummer theory. We also study in detail the error term of the asymptotics and show that it is $O(X^{\alpha})$, for an explicit $\alpha<1$.Comment: 19 page

A dissociation between real and simulated movements in Parkinson's disease

Subcortical lesions have been simultaneously implicated in both real and simulated movement deficits. However, the analysis of the simulated opposition axis in precision grasping reveals that, in individuals with idiopathic bilateral Parkinson's disease motor imagery is impaired and that execution of overt movements is spared. This constitutes the first lesion observation congruent with the anatomical and functional dichotomy between real and simulated movements seen in experimental studies. These results underline the modality-specific nature of motor imagery and show that subcortical damage differentially impacts on motor activity

Finite hypergeometric functions

Finite hypergeometric functions are complex valued functions on finite fields which are the analogue of the classical analytic hypergeometric functions. From the work of N.M.Katz it follows that their values are traces of Frobenius on certain l-adic sheafs. More concretely, in many instances their values can be used to give formulas for pointcounts of F_q-rational points on certain varieties. In this paper we work out the case of one-variable functions whose monodromy in the analytic case can be defined over the rational integers.Comment: 26 pages, 2 figure

Ap\'ery Acceleration of Continued Fractions

We explain in detail how to accelerate continued fractions (for constants as well as for functions) using the method used by R.~Ap\'ery in his proof of the irrationality of $\zeta(3)$. We show in particular that this can be applied to a large number of continued fractions which can be found in the literature, thus providing a large number of new continued fractions. As examples, we give a new continued fraction for $\log(2)$ and for $\zeta(3)$, as well as a simple proof of one due to Ramanujan